Re: Fourth nerve palsy and double vision
This really depends on the measurements of your congenital fourth nerve palsy in terms of surgical correction. For some minor congenital fourth nerve palsies the reason the eye drifts up is because the inferior oblique muscle in the eye affected with the palsy acts unopposed by the 'weak' superior oblique muscle (which is the muscle supplied by the fourth nerve, an alternate name for your condition being congenital superior oblique palsy) . Hence by 'weakening' the inferior oblique muscle by myectomy (they cut a bit out) or disinsertion from the globe of the eye (the muscle reattaches itself further back) this could reduce the deviation in any meaningful gaze to the point of being considered a cure.
This type of surgery, if you were a suitable candidate, is relatively safe and effective. Occasionally with these surgeries the condition is undercorrected and another eye muscle may need to be operated on. However, overcorrection is not a significant risk, which is unlike many types of eye muscle surgery where it is a prime reason not to operate. If you overcorrect a deviation then you have double vision in the opposite direction your brain can't control = bad. You might think operating on the affected superior oblique muscle is the sensible thing to do, but surgically it is not and carries far more risks. As your fourth nerve palsy is congenital it may be more to do with a mechanical dysfunction of the superior oblique tendon rather than an actual palsy of the fourth cranial nerve.
So it all depends on the measurements of your palsy, and the specialist you see. Not all ophthalmologists (eye doctors) are the same and have the same experience or expertise. It may be worth getting a second or third opinion. Check out the credentials of the ophthalmologists you see to ensure they specialise in strabismus (a strabismologist). You have a cyclovertical strabismus/squint/eye muscle imbalance.
They may also consider if the double vision or head tilting (something most people with congenital fourth nerve palsy do to reduce their double vision and may be unaware of) is becoming a problem. So if your double vision is becoming more frequent or the head tilting is causing neck pain these may be considered reasons to operate. And these symptoms generally worsen with age.
In so far as the military are concerned I do not know if they would accept surgical correction.
If your condition were not affecting you, you wouldn't be getting double vision and the military would let you in.